Avian Influenza, with a
high mortality rate in human population, is considered to be one of the most
significant potential threats to human beings. Based on a recent avian
influenza SI-SIR model with logistic growth for birds, we propose a stochastic
model with generalized incidence rate. For the stochastic avian-only system,
sufficient conditions for the extinction of infected birds are established, and
the existence of a unique ergodic stationary distribution is also obtained. For
the stochastic avian-human system, a threshold number ？is established, and
hence the extinction of disease is investigated. From the viewpoint of biology,
the noise intensity in the infected birdsplays
a key role in the evolutionary dynamics. Moreover, we also analyze the
asymptotic behavior around the endemic equilibrium of the corresponding
deterministic model.
References
[1]
Liu, S., Ruan, S. and Zhang, X. (2017) Nonlinear Dynamics of Avian Influenza Epidemic Models. Mathematical Biosciences, 283, 118-135.
https://doi.org/10.1016/j.mbs.2016.11.014
[2]
Centers for Disease Control and Prevention (CDC) (2017) First Human Infection with Avian Influenza A(H5N1) Virus Since Sept. Reported in Nepal.
https://www.cdc.gov/flu/spotlights/2018-2019/h5n1-human-infection.html
[3]
Alexander, D.J., Allan, W.H., Parsons, D. and Parsons, G. (1978) The Pathogenicity of Four Avian Influenza Viruses for Chickens, Turkeys and Ducks. Research in Veterinary Science, 24, 242-247. https://doi.org/10.1016/S0034-5288(18)33080-7
[4]
Grais, R.F., Ellis, J.H. and Glass, G.E. (2003) Assessing the Impact of Airline Travel on the Geographic Spread of Pandemic Influenza. European Journal of Epidemiology, 18, 1065-1072. https://doi.org/10.1023/A:1026140019146
[5]
Ferguson, N.M. and Cummings, D.A.T. (2005) Strategies for Containing an Emerging Influenza Pandemic in Southeast Asia. Nature, 437, 209-214.
https://doi.org/10.1038/nature04017
[6]
Menach, A.L. (2006) Key Strategies for Reducing Spread of Avian Influenza among Commercial Poultry Holdings: Lessons for Transmission to Humans. Proceedings of the Royal Society B, 273, 2467-2475. https://doi.org/10.1098/rspb.2006.3609
[7]
Shi, Z., Zhang, X. and Jiang, D. (2019) Dynamics of an Avian Influenza Model with Half-Saturated Incidence. Applied Mathematics and Computation, 355, 399-416.
https://doi.org/10.1016/j.amc.2019.02.070
[8]
Iwami, S., Takeuchi, Y. and Liu, X. (2007) Avian-Human Influenza Epidemic Model. Mathematical Biosciences, 207, 1-25.
https://doi.org/10.1016/j.mbs.2006.08.001
[9]
Liu, S., Pang, L., Ruan, S. and Zhang, X. (2015) Global Dynamics of Avian Influenza Epidemic Models with Psychological Effect. Computational and Mathematical Methods in Medicine, 2015, 1-12. https://doi.org/10.1155/2015/913726
[10]
Tuncer, N. and Martcheva, M. (2013) Modeling Seasonality in Avian Influenza H5N1. Journal of Biological Systems, 21, 1-30.
https://doi.org/10.1142/S0218339013400044
[11]
Liu, M., Huang, J., Ruan, S. and Yu, P. (2019) Bifurcation Analysis of an SIRS Epidemic Model with a Generalized Nonmonotone and Saturated Incidence Rate. Journal of Differential Equations, 267, 1859-1898.
https://doi.org/10.1016/j.jde.2019.03.005
[12]
Zhang, Y., Zhang, L. and Yuan, S. (2018) The Effect of Media Coverage on Threshold Dynamics for a Stochastic SIS Epidemic Model. Physica A, 512, 248-260.
https://doi.org/10.1016/j.physa.2018.08.113
[13]
Liu, Q. and Jiang, D. (2017) Stationary Distribution and Extinction of a Stochastic SIR Model with Nonlinear Perturbation. Applied Mathematics Letters, 73, 8-15.
https://doi.org/10.1016/j.aml.2017.04.021
[14]
Meng, X., Zhao, S. and Feng, T. (2016) Dynamic of a Novel Nonlinear Stochastic SIS Epidemic Model with Double Epidemic Hypothesis. Journal of Mathematical Analysis and Applications, 433, 227-242. https://doi.org/10.1016/j.jmaa.2015.07.056
[15]
Li, D., Cui, J., Liu, M. and Liu, S. (2015) The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate. Bulletin of Mathematical Biology, 77, 1705-1743. https://doi.org/10.1007/s11538-015-0101-9
[16]
Zhang, F. and Zhang, X. (2018) The Threshold of a Stochastic Avian-Human Influenza Epidemic Model with Psychological Effect. Physica A, 492, 485-495.
https://doi.org/10.1016/j.physa.2017.10.043
[17]
Zhang, X. (2017) Global Dynamics of a Stochastic Avian Chuman inluenza Epidemic Model with Logistic Growth for Avian Population. Nonlinear Dynamics, 90, 2331-2343. https://doi.org/10.1007/s11071-017-3806-5
Ikeda, N. and Watanabe, S. (1997) A Comparison Theorem for Solutions Of Stochastic Differential Equations and Its Applications. Osaka. Journal of Mathematics, 14, 619-633.
[20]
Has’minskii, R.Z. (1980) Stochastic Stability of Differential Equations. Sijthoff & Noordhoff. Alphen aan den Rijn, The Netherlands.
[21]
Kushner, H.J. (1967) Stochastic Stability and Control. Academic Press, New York.
